math and vote Vote! Why your ballot
isn't as meaningless as you think.
By Jordan Ellenberg Posted Monday, Oct. 11, 2004, at
Let's say, for the sake of argument, you were among the 72,000 people who
participated in the Guinness-certified world's largest chicken dance in Canfield, Ohio,
in 1996. You probably feel pretty proud. But according to Slate's
Stephen E. Landsburg, you shouldn't. After all,
unless a previous chicken dance for 71,999 were on the books, your
participation made no difference; the record would have fallen whether or not
you'd shown up.
Landsburg is arguing against voting, not
chicken-dancing: Your presidential vote, he says, "will never matter
unless the election in your state is within one vote of a dead-even tie."
That, of course, is extremely unlikely. So, the negligible chance of casting
the deciding ballot is outweighed by the small but certain costs of voting,
like the gas you'll use and the time you'll spend.
And yet people vote anyway, by the millions. Political scientists call this
conundrum "the paradox of voting," and you could stay up half the night
(I just did) reading research literature on the subject. Why do people vote
when it's so unlikely to matter? Maybe because the
pleasurable feeling of doing one's duty offsets the cost of gas.Maybe because people have an interest in their candidate not just
winning but winning by as large a margin as possible. Maybe because
we're motivated to avoid even small possibilities of regret—the regret that
those Al Gore supporters who sat out Florida
in 2000 surely feel, whether economists think they're being rational or not.
But let's stick to mathematics. Suppose we grant to Landsburg
that voting carries a certain cost and that your vote should be considered
worthwhile only if it decides the election. Everyone can agree that's
unlikely—but how unlikely? Landsburg first
modeling voters in a state, say Florida,
as 6,000,000 coin-flippers, each choosing George Bush with some probability p
and John Kerry with probability 1-p. For instance, if p is
1/2, 1-p is also 1/2; each voter has an equal chance of selecting Bush
or Kerry. As you might expect, the odds of a tied outcome are not bad—about 1
in 3,100, as Landsburgcomputes.
But p might not be 1/2, and even a tiny bias in voter preference
can make a tie exceedingly unlikely. For instance, if p = .51, the
chance of a tie drops to 1 in 101046, a probability so small as to be effectively zero. (Here's Landsburg's
computation.) Your vote is not going to count.
So, are we back to Landsburg's discouraging
conclusion that voting is most often a waste of time? Not quite, because it's
impossible to know in advance what proportion of your fellow Floridians are
planning to vote for Bush. If you knew p was exactly 1/2, you'd be
sure to get out and vote. If you knew Bush held a 51 percent advantage, you'd
be foolish to bother. But you don't know, and without that knowledge you can't
reason as Landsburg wants you to.
You don't know, but you can guess. A Sept. 29 poll of 704 Florida
voters by CNN/USA has Bush leading Kerry 52-43. For simplicity, let's dump the
still-undecideds and third-party enthusiasts and say
that, among 669 randomly selected likely Florida
voters, 366 supported Bush and 303 Kerry, a 55-45 margin in Bush's favor. If
forced to make a guess, we might expect 55 percent of Florida
voters to favor Bush. But how confident should we be that our guess is right?
In particular, how likely is it that the real proportion of Bush votes in the
state is very close to 50 percent?
The inconvenient truth is that the poll alone can't tell you. If, for
instance, a poll in Massachusetts
showed a 10-point Bush lead, we'd still think Bush was behind, though we might
rate the race closer than we did previously. Our best guess about the true
state of things represents a compromise between our prior intuitions and the
The mathematical method by which this compromise is hammered out is called
Bayesian inference. The computations involved, though elementary, are a bit
tedious to include here, but stats fans can find more in the accompanying
computations page. Let's suppose we start out with the (somewhat
unrealistic) belief that the true vote count for Bush in Florida
is equally likely to be any number between zero and 6,000,000. Given the 52-43
poll result, the Bayesian computation puts the chance of a tie at about 1 in 5
million. If the polls were exactly even, the chance would go up to 1 in
300,000. Those still aren't fantastic odds, but both beat the 1-in-80 million
chance of winning Powerball by a mile. Suddenly
voting seems a lot more justifiable.
Even if your vote helps swing Florida, Florida
might not swing the election. But if the electoral vote is sufficiently close,
many states could be in a position to affect the national outcome. You know
that if 538 fewer Bush votes had been counted in Florida,
Al Gore would be president. But did you know that only 1,231,944 more Bob Dole
voters, carefully apportioned among Nevada, Kentucky, Arizona, Tennessee, New
Mexico, Florida, New Hampshire, Delaware, Ohio, and Pennsylvania, would have
given their man the election, despite Clinton's lead of 8 million in the
It's precisely this sensitivity to small swings in key states that makes people fume about
the Electoral College—that saturates Tampa
with campaign ads and volunteers and leaves Los Angelesquiet, that makes elections vulnerable to targeted
fraud beforehand and targeted lawsuits afterwards. So, let's take a moment to
cheer this one fine feature of our system: It puts many voters in many states
on notice that their vote might really count. The state that swings
could be your own. So, ignore Landsburg! Take your
place in this big majestic chicken dance we call democracy! Vote!
Thanks to John Londregan and Howard Rosenthal
for helpful suggestions and pointers to relevant literature.